Re: [Quantlib-users] The reason of differences between `FDDividendAmericanEngine` of QuantLib 1.12 and `FdBlackScholesVanillaEngine` of QuantLib 1.18

[Klaus S. <kl...@sp...>]

Hi,

thanks for sharing your results. 

W.r.t. your questions
1. Douglas and Crank-Nicolson scheme are exactly the same for one-dimensional problems like the  Black-Scholes equation. In higher dimensions operator splitting schemes like Douglas or Craig-Sneyd are better suited than Crank-Nicolson, hence there wasn't really a need to implement it (even though for the sake of completeness the Crank-Nicolson scheme has been added with the 1.18 release in C++).
2. The spot dividend model is described e.q. in "Back to Basics: a new approach to the discrete dividend problem". People might argue that the spot dividend process is more realistic for path dependent options like barrier options or structured equity notes.
3.  As far as I know there is no general and meaningful algorithm for this problem, which is as easy as for Monte-Carlo.

Observation 1 on Case 2. If I remember correctly then there was subtle difference between interest rate treatment for discrete dividents between the FDDividendAmericanEngine on the one side and AnalyticDividendEuropeanEngine,FdBlackScholesVanillaEngine on the other side.
Observation 2: For these pathologic cases the default number of S grid steps is too small. In general I'd use 500 or more S grid steps.

best regards
Klaus